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Multilevel Converters Analysis, Modulation, Topologies, and Applications Edited by Gabriele Grandi and Alex Ruderman Printed Edition of the Special Issue Published in Energies www.mdpi.com/journal/energies Multilevel Converters Multilevel Converters: Analysis, Modulation, Topologies, and Applications Special Issue Editors Gabriele Grandi Alex Ruderman MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Gabriele Grandi Alex Ruderman University of Bologna Nazarbayev University Italy Kazakhstan Editorial Ofﬁce MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Energies (ISSN 1996-1073) from 2018 to 2019 (available at: https://www.mdpi.com/journal/energies/special issues/multilevel converters) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03921-481-5 (Pbk) ISBN 978-3-03921-482-2 (PDF) c 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface to ”Multilevel Converters: Analysis, Modulation, Topologies, and Applications” . . . xi Jin Zhu, Tongzhen Wei, Qunhai Huo and Jingyuan Yin A Full-bridge Director Switches based Multilevel Converter with DC Fault Blocking Capability and Its Predictive Control Strategy Reprinted from: energies 2019, 12, 91, doi:10.3390/en12010091 . . . . . . . . . . . . . . . . . . . . 1 Kyoung-Pil Kang, Younghoon Cho, Myung-Hyo Ryu and Ju-Won Baek A Harmonic Voltage Injection Based DC-Link Imbalance Compensation Technique for Single-Phase Three-Level Neutral-Point-Clamped (NPC) Inverters Reprinted from: energies 2018, 11, 1886, doi:10.3390/en11071886 . . . . . . . . . . . . . . . . . . . 23 Eun-Su Jun and Sangshin Kwak A Highly Efﬁcient Single-Phase Three-Level Neutral Point Clamped (NPC) Converter Based on Predictive Control with Reduced Number of Commutations Reprinted from: energies 2018, 11, 3524, doi:10.3390/en11123524 . . . . . . . . . . . . . . . . . . . 38 Ming Wu, Zhenhao Song, Zhipeng Lv, Kai Zhou and Qi Cui A Method for the Simultaneous Suppression of DC Capacitor Fluctuations and Common-Mode Voltage in a Five-Level NPC/H Bridge Inverter Reprinted from: energies 2019, 12, 779, doi:10.3390/en12050779 . . . . . . . . . . . . . . . . . . . . 66 Jingyuan Yin, Wen Wu, Tongzhen Wei, Xuezhi Wu and Qunhai Huo A Novel Fault-Tolerant Control of Modular Multilevel Converter under Sub-Module Faults Based on Phase Disposition PWM Reprinted from: energies 2019, 12, 20, doi:10.3390/en12010020 . . . . . . . . . . . . . . . . . . . . 80 Rafael S. Leite, João L. Afonso and Vı́tor Monteiro A Novel Multilevel Bidirectional Topology for On-Board EV Battery Chargers in Smart Grids Reprinted from: energies 2018, 11, 3453, doi:10.3390/en11123453 . . . . . . . . . . . . . . . . . . . 97 Yungdeug Son and Jangmok Kim A Novel Phase Current Reconstruction Method for a Three-Level Neutral Point Clamped Inverter (NPCI) with a Neutral Shunt Resistor Reprinted from: energies 2018, 11, 2616, doi:10.3390/en11102616 . . . . . . . . . . . . . . . . . . . 118 Zongbin Ye, Anni Chen, Shiqi Mao, Tingting Wang, Dongsheng Yu and Xianming Deng A Novel Three-Level Voltage Source Converter for AC–DC–AC Conversion Reprinted from: energies 2018, 11, 1147, doi:10.3390/en11051147 . . . . . . . . . . . . . . . . . . . 136 Weide Guan, Shoudao Huang, Derong Luo and Fei Rong A Reverse Model Predictive Control Strategy for a Modular Multilevel Converter Reprinted from: energies 2019, 12, 297, doi:10.3390/en12020297 . . . . . . . . . . . . . . . . . . . . 153 Van-Quang-Binh Ngo, Minh-Khai Nguyen, Tan-Tai Tran, Young-Cheol Lim and Joon-Ho Choi A Simpliﬁed Model Predictive Control for T-Type Inverter with Output LC Filter Reprinted from: energies 2019, 12, 31, doi:10.3390/en12010031 . . . . . . . . . . . . . . . . . . . . 168 v Salvatore Foti, Giacomo Scelba, Antonio Testa and Angelo Sciacca An Averaged-Value Model of an Asymmetrical Hybrid Multi-Level Rectiﬁer Reprinted from: energies 2019, 12, 589, doi:10.3390/en12040589 . . . . . . . . . . . . . . . . . . . . 186 Zheng Gong, Qi Cui, Xi Zheng, Peng Dai and Rongwu Zhu An Improved Imperialist Competitive Algorithm to Solve the Selected Harmonic Elimination Pulse-Width Modulation in Multilevel Converters Reprinted from: energies 2018, 11, 3080, doi:10.3390/en11113080 . . . . . . . . . . . . . . . . . . . 205 Jianfei Chen, Caisheng Wang and Jian Li An Input-Parallel-Output-Series Switched-Capacitor Three-level Boost Converter with a Three-Loop Control Strategy Reprinted from: energies 2018, 11, 2631, doi:10.3390/en11102631 . . . . . . . . . . . . . . . . . . . 221 Manel Hammami, Gabriele Rizzoli, Riccardo Mandrioli and Gabriele Grandi Capacitors Voltage Switching Ripple in Three-Phase Three-Level Neutral Point Clamped Inverters with Self-Balancing Carrier-Based Modulation Reprinted from: energies 2018, 11, 3244, doi:10.3390/en11123244 . . . . . . . . . . . . . . . . . . . 247 Sridhar Vavilapalli, Umashankar Subramaniam, Sanjeevikumar Padmanaban and Frede Blaabjerg Design and Controller-In-Loop Simulations of a Low Cost Two-Stage PV-Simulator Reprinted from: energies 2018, 11, 2774, doi:10.3390/en11102774 . . . . . . . . . . . . . . . . . . . 267 Fabio Viola Experimental Evaluation of the Performance of a Three-Phase Five-Level Cascaded H-Bridge Inverter by Means FPGA-Based Control Board for Grid Connected Applications Reprinted from: energies 2018, 11, 3298, doi:10.3390/en11123298 . . . . . . . . . . . . . . . . . . . 282 Bin Jiang, Yanfeng Gong and Yan Li Fault Detection and Location of IGBT Short-Circuit Failure in Modular Multilevel Converters Reprinted from: energies 2018, 11, 1492, doi:10.3390/en11061492 . . . . . . . . . . . . . . . . . . . 329 Joan Nicolas-Apruzzese, Emili Lupon, Sergio Busquets-Monge, Alfonso Conesa, Josep Bordonau and Gabriel Garcı́a-Rojas FPGA-Based Controller for a Permanent-Magnet Synchronous Motor Drive Based on a Four-Level Active-Clamped DC-AC Converter Reprinted from: energies 2018, 11, 2639, doi:10.3390/en11102639 . . . . . . . . . . . . . . . . . . . 342 Mattia Ricco, Laszlo Mathe, Eric Monmasson and Remus Teodorescu FPGA-Based Implementation of MMC Control Based on Sorting Networks Reprinted from: energies 2018, 11, 2394, doi:10.3390/en11092394 . . . . . . . . . . . . . . . . . . . 359 Zhi Wu, Jiawei Chu, Wei Gu, Qiang Huang, Liang Chen and Xiaodong Yuan Hybrid Modulated Model Predictive Control in a Modular Multilevel Converter for Multi-Terminal Direct Current Systems Reprinted from: energies 2018, 11, 1861, doi:10.3390/en11071861 . . . . . . . . . . . . . . . . . . . 377 Jiazheng Lu, Siguo Zhu, Bo Li, Yanjun Tan, Xiudong Zhou, Qinjun Huang, Yuan Zhu and Xinguo Mao Low-Harmonic DC Ice-Melting Device Capable of Simultaneous Reactive Power Compensation Reprinted from: energies 2018, 11, 2596, doi:10.3390/en11102596 . . . . . . . . . . . . . . . . . . . 394 vi Yajun Ma, Hua Lin, Zhe Wang and Zuyao Ze Modiﬁed State-of-Charge Balancing Control of Modular Multilevel Converter with Integrated Battery Energy Storage System Reprinted from: energies 2019, 12, 96, doi:10.3390/en12010096 . . . . . . . . . . . . . . . . . . . . 411 Salvatore Foti, Antonio Testa, Salvatore De Caro, Tommaso Scimone, Giacomo Scelba and Giuseppe Scarcella Multi-Level Open End Windings Multi-Motor Drives Reprinted from: energies 2019, 12, 861, doi:10.3390/en12050861 . . . . . . . . . . . . . . . . . . . . 431 Zhansen Akhmetov, Manel Hammami, Gabriele Grandi and Alex Ruderman On PWM Strategies and Current THD for Single- and Three-Phase Cascade H-Bridge Inverters with Non-Equal DC Sources Reprinted from: energies 2019, 12, 441, doi:10.3390/en12030441 . . . . . . . . . . . . . . . . . . . . 450 Xiaoqiong He, Haijun Ren, Jingying Lin, Pengcheng Han, Yi Wang, Xu Peng and Zeliang Shu Power Flow Analysis of the Advanced Co-Phase Traction Power Supply System Reprinted from: energies 2019, 12, 754, doi:10.3390/en12040754 . . . . . . . . . . . . . . . . . . . . 467 Elie Talon Louokdom, Serge Gavin, Daniel Siemaszko, Frédéric Biya-Motto, Bernard Essimbi Zobo, Mario Marchesoni and Mauro Carpita Small-Scale Modular Multilevel Converter for Multi-Terminal DC Networks Applications: System Control Validation Reprinted from: energies 2018, 11, 1690, doi:10.3390/en11071690 . . . . . . . . . . . . . . . . . . . 487 Peter Zajec and Mitja Nemec Theoretical and Experimental Investigation of the Voltage Ripple across Flying Capacitors in the Interleaved Buck Converter with Extended Duty Cycle Reprinted from: energies 2018, 11, 1017, doi:10.3390/en11041017 . . . . . . . . . . . . . . . . . . . 506 Driss Oulad-Abbou, Said Doubabi and Ahmed Rachid Voltage Balance Control Analysis of Three-Level Boost DC-DC Converters: Theoretical Analysis and DSP-Based Real Time Implementation Reprinted from: energies 2018, 11, 3073, doi:10.3390/en11113073 . . . . . . . . . . . . . . . . . . . 519 vii About the Special Issue Editors Gabriele Grandi received his M.Sc. (cum laude) and Ph.D. degrees in Electrical Engineering from the University of Bologna, Bologna, Italy, in 1990 and 1994, respectively. He has been with the Department of Electrical, Electronic, and Information Engineering, University of Bologna as Research Associate (since 1995), Associate Professor (since 2005), and most recently as Full Professor (since 2016) in Electrical Engineering. He is the Founder and Leader of the research laboratory “SolarTronic-Lab” at University of Bologna, dealing with power electronic circuits, multiphase and multilevel converters, photovoltaics, electric vehicle chargers, and circuit modeling. He has authored or co-authored more than 160 papers in conference proceedings and international journals, mainly with the IEEE. Dr. Grandi serves as Editor-at-Large for IET Power Electronics, Academic Editor for MDPI Energies and MDPI Electronics, and Associate Editor for IEEE Trans. on Industrial Electronics. Alex Ruderman obtained his M.Sc. (cum laude) and Ph.D. degrees from the Electrotechnical University and Polytechnic University (former Leningrad, USSR; now St. Petersburg, Russia) in 1980 and 1987 respectively. In 1995–2003, he worked as an R&D engineer for Intel Microprocessor Development Center, Haifa, Israel. In 2004–2013, Alex taught electronics-related courses in Bar Ilan University, Ariel University and Holon Institute of Technology as Adjunct Professor. In 2006, he joined Elmo Motion Control, Petach Tikva, Israel, the makers of compact intelligent servo drives, as a Chief Scientist (Elmo drives are used by NASA in Mars Curiosity and InSight missions). Since 2013, Alex has been Associate Professor at Nazarbayev University School of Engineering and Digital Sciences, Electrical and Computer Engineering Department. Alex is a regular reviewer for IEEE Transactions on Industrial Electronics and Power Electronics and a program committee member for several international Power Electronics Conferences. His major research focus is multilevel power converters—he has authored over 60 conference and journal papers on the subject. Alex is an IEEE Senior Member and Associate Editor for the IET Journal of Power Electronics. ix Preface to ”Multilevel Converters: Analysis, Modulation, Topologies, and Applications” Multilevel inverters (MLIs) have been widely used for medium- and high-voltage power applications in the recent decades, mainly for grid-connected applications to interface with renewable energy sources. Compared to basic two-level inverters, MLIs offer many advantages, such as reduced voltage rating of power switches, reduced voltage and current harmonic distortion, reduced electromagnetic interference, as well as ﬂexibility and modularity. MLIs became a standard for applications such as in medium voltage drives and HVDC grids, and are promising for lower voltage applications such as in battery chargers, active ﬁlters, static compensators, dynamic voltage restorers, rectiﬁers, grid-tied inverters, and many more. Increased efﬁciency and reduced harmonic distortion are beneﬁcial for photovoltaic systems and uninterruptible power supplies. The introduction of multilevel topologies has shifted the power converter design paradigm, including control and modulation strategies, component selection and requirements, reliability aspects, amongst others. While relatively low-power applications employ high-frequency PWM, for high-voltage/current applications, the switching frequency of the power semiconductors is limited to few kHz by switching loss considerations, and the use of multilevel converters becomes mandatory. Gabriele Grandi, Alex Ruderman Special Issue Editors xi energies Article A Full-bridge Director Switches based Multilevel Converter with DC Fault Blocking Capability and Its Predictive Control Strategy Jin Zhu, Tongzhen Wei *, Qunhai Huo and Jingyuan Yin Institute of Electrical Engineering, Chinese Academy of Sciences, Haidian District, Beijing 100190, China; [email protected] (J.Z.); [email protected] (Q.H.); [email protected] (J.Y.) * Correspondence: [email protected]; Tel.: +86-182-1116-1108 Received: 23 October 2018; Accepted: 21 December 2018; Published: 28 December 2018 Abstract: Voltage source converter-based high-voltage direct current transmission system (VSC-HVDC) technology has been widely used. However, traditional half-bridge sub module (HBSM)-based module multilevel converter (MMC) cannot block a DC fault current. This paper proposes that a full-bridge director switches based multi-level converter can offer features such as DC side fault blocking capability and is more compact and lower cost than other existing MMC topologies. A suitable predictive control strategy is proposed to minimize the error of the output AC current and the capacitor voltage of the sub-module while the director switches are operated in low-frequency mode. The validity of the proposed topology and control method is demonstrated based on simulation and experimental studies. Keywords: multilevel converter; DC side fault blocking; predictive control 1. Introduction The modular multilevel converter (MMC) has been accepted as a suitable solution for high-voltage and high-power application ﬁelds due to several inherent features [1–8]. However, blocking the DC fault current becomes a difﬁcult problem because the anti-parallel diodes are still conducting after the insulated-gate bipolar transistors (IGBTs) of HBSM are turned off [9]. To solve this problem, recent research has highlighted a number of interesting converter topologies which combine the features of the multilevel output AC voltage waveform and DC fault blocking capacity [9–22]. Full-bridge sub-module (FBSM) based MMC (F-MMC) is a basic conﬁguration with DC fault current blocking capacity [10,11]. However, the DC fault current blocking capability comes at a cost of nearly doubling power losses and number of semiconductor devices. Some other type of sub-module is proposed instead of FBSM to make a further optimization in reducing the number of IGBTs, such as a clamp double sub-module (CDSM) proposed in [14,15] and a three-level cross-connected sub-module (TCSM) proposed in [16]. Several hybrid MMC topologies are also proposed, based on HBSM and those various types of sub-module [9,12,13,16–19,23], for further reducing the cost and loss on the premise of having the DC fault blocking capacity, such as hybrid MMC based on CDSM and HBSM (CH-MMC). However, there are still some drawbacks, for example, as they are composed of a large number of sub-modules, the system needs to be more complicated and the converter station bulkier. The alternate-arm multilevel converter (AAMC) based on the hybrid topology of HBSM and director switches is proposed in [20–22]. The AAMC further improves the traditional MMC topology by cutting the number of sub-modules, reducing DC bus voltage, and gaining the ability to block DC fault currents [22]. However, some features still have the possibility for further optimization, such as the size and cost of the overall system. One of the main technical challenges Energies 2019, 12, 91; doi:10.3390/en12010091 1 www.mdpi.com/journal/energies Energies 2019, 12, 91 associated with the control of such a director switches based multilevel converter is to simultaneously keep the capacitor voltages balanced and provide good output current tracking performance, while the director switches keep switching in low frequency. In order to further optimize the size and cost of the voltage source converter-based high-voltage direct current transmission system (VSC-HVDC) converter with the blocking ability of DC faults, this paper proposes a full bridge director switches based alternate-arm multi-level converter (FA-MMC) and a corresponding control strategy: 1. The size and cost of the overall system can be signiﬁcantly reduced by reducing the number of SM capacitors, IGBTs, and other related devices. In addition, an FA-MMC retains the ability to block DC-side faults since it uses H-bridge SMs as the AAMC. 2. Similar to AAMC, a systematic multi-objective control method is needed for this kind of topology to minimize the error of output AC current and the capacitor voltage of the sub-module while the director switches are operated in low-frequency mode. A suitable predictive control strategy for this kind of topology is presented in this paper to achieve the ﬂexibility to include the previously mentioned multiple system requirements. 2. Proposed Topology 2.1. Structure and Basic Operation The basic circuit conﬁguration of full-bridge director switches based modular multi-Level converter (FA-MMC) proposed in this paper is shown in Figure 1b. The proposed topology consists of a stack of H-bridge SMs and four director switches (S1–S4) made of series IGBTs or IGCTs. The ability of DC-side fault blocking is still retained since the H-bridge SMs structure is the same as the AAMC. Pÿ Vdc/2 P Upper stack of Cell H-bridge cells SiVci Vupper Cell C Vci Cell Vstack Stack of H-bridge cells t Cell Buffer inductor t Director switch B Vdirector Vac O Vac ib Buffer inductor UDC t M t t Vlower S1 S2 Director t Cell switch A C R Ls Lower stack of is H-bridge cells Cell -Vdc/2 S3 S4 Nÿ N (a) (b) Figure 1. Schematic representation of the two topologies: (a) alternate-arm multilevel converter (AAMC); (b) full bridge director switches based alternate-arm multi-level converter (FA-MMC). The voltage of the director switches (Udirector in Figure 1b) is equal to the DC voltage (UDC in Figure 2) plus the voltage produced by the stack of H-bridge SMs which can be considered as only one controllable voltage source. Therefore, the voltage of director switches can be adjusted ﬂexibly so that the switching of S1 –S4 can switch at near to zero voltage. The ideal voltage waveform is shown in Figure 2a. 2 Energies 2019, 12, 91 (a) (b) Figure 2. The voltage waveform and state of S1–S4: (a) without energy balance mode; (b) with energy balance mode. The working cycle of S1 –S4 is synchronized with the output AC voltage. S1 and S4 are conducting and S2 and S3 are turned off while the output AC voltage (Uac in Figure 1b is in its positive half-cycle, in contrast, S2 and S3 are conducting and S1 and S4 are turned off while the output AC voltage is in its negative half-cycle. This ensures that the four director switches can switch at low-frequency and at the point of zero-voltage-crossing as shown in Figure 2a. These features lead to low switching losses, and low demand for dynamic voltage sharing at the switching instant of the series switches, so that the system design has been simpliﬁed. 2.2. Energy Balance When the AC current ﬂows through the stack of H-bridge sub module. In order to ensure the continuous operation of the system, the energy balance of the stack of H-bridges should be guaranteed. The amount of energy transferred from the AC side (EAC ) and going to the DC side (EDC ) should be equal over half the fundamental period and is given as ∧ ∧ 3 V AC I AC E AC = π cos( ϕ), (1) 2 ω ∧ 6UDC I AC EDC = cos( ϕ), (2) ω For EAC to equal EDC , the relationship between the DC voltage magnitudes and AC voltage magnitudes mentioned in Equations (1) and (2) can be given as π∧ UDC = V , (3) 4 AC However, since the converters can’t operate in the perfect given by Equation (3), an energy balancing strategy should be used. Reference [22] presented two methods to achieve energy balance for AAMCs that can also be used in this topology: Overlap current and third harmonic current injection. In this paper the overlap current method is used to extend the period when the current directed from S1 and S4 to S2 and S3 is extended and S1 –S4 are all conducting. The overlap current is used to exchange power between 3 Energies 2019, 12, 91 the sub module capacitors and the DC bus. The load current is only slightly affected, since the overlap time is very short and the inductance can smooth the change in current. Considering its effect on the grid current, the overlap time is determined to be less than 0.8 ms. 3. Predictive Control Strategy The control strategy of the proposed topology requires minimizing the error of the output current and DC voltage in each sub module, and, meanwhile, the director switches switching should be operated in low-frequency and zero-voltage switching mode. 3.1. Dynamic Modeling Based on Figure 2, the governing equations of the single-phase FA-MMC can be shown as follows: dib UDC − VPB − Lb = VMN (4) dt dis VAC = Ls + Ris (5) dt VAC = Sd VMN (6) As presented in Section 2, the value of Lb is small and the voltage on it can be ignored; S1 –S4 have ﬁve switching state combinations depending on a switching function Sd as shown in Table 1. Table 1. Switching states of director switches. Mode Sd S1 S2 S3 S4 Output Voltage (VAC ) 1 ON OFF OFF ON VMN Basic Operating Mode −1 OFF ON ON OFF −VMN 0 ON ON ON ON 0 Energy Balancing Mode 0 ON ON OFF OFF 0 0 OFF OFF ON ON 0 The output voltage of each H-bridge sub module is equal to Vci (capacitor voltage of the ith sub module (i = 1, 2, · · · , n)), −Vci , or zero, depending on the switching states, and depends on a switching function Si ⎧ ⎪ ⎨ 1 Si = 0 (i = 1, 2, . . . , n). (7) ⎪ ⎩ −1 The relationship between Vci and UDC is formalized as n ∑ Vci ≈ UDC (8) i =1 Based on Equation (16) and the basic principle, VPB is formalized as n ∑ Si Vci = VPB (9) i =1 The dynamic capacitor voltage of the cells of the H-bridge sub module in Figure 1b is formalized as dVci Si i b = C (10) dt 4 Energies 2019, 12, 91 The relationship of currents is and ib in Figure 1b, which was also indicated by the switching function Sd according to Table 1, is expressed as ib = Sd is (Sd = 1 or − 1) (11a) n dib L = UDC − ∑ Si Vci (Sd = 0) (11b) dt i =1 The switching states of director switches operate in an energy balancing mode, as mentioned in Table 1. As discussed previously, the current ib ﬂows through the stack of H-bridge sub modules, buffer inductor, and director switch to the DC side, charging or discharging the capacitor of the H-bridge sub modules. Only considering the basic operating mode, substituting Equations (5), (6), (9), and (11a) into (4), a dynamic model of the single-phase proposed topology in basic operating mode can be expressed as n dis dis Sd (UDC − ∑ Si Vci − Sd Lb ) = Ls + Ris (12) i =0 dt dt where Sd = 1 or −1. Equation (12) can be simpliﬁed as n dis L = Sd (UDC − ∑ Si Vci ) − Ris (13) dt i =0 where L = Lb + Ls . 3.2. Proposed Predictive Control The predictive control strategy is proposed in this section based on the dynamic model of the FA-MMC presented above, the three primary targets of the predictive control strategy is achieved as follows: 3.2.1. AC-Side Current Control Assuming a sampling period of Ts , a discrete-time model of the FA-MMC AC-side current in basic operating mode based on Equation (3) is calculated by n L (is (k + 1) − is (k)) = Sd (k)(UDC (k) − ∑ Si (k)Vci (k)) − Ris (k) (14) Ts i =0 the value of Sd could be assumed as a constant value during a short sampling period of Ts . is (k) is the actual AC current at time k and is (k + 1) is the predicted AC current at time k + 1, UDC (k) can be considered as a constant value if the DC side voltage is controlled. Finally, Vci (k) is the capacitor voltage of the sub module i at time k. To reduce the error between the predicted current and the reference current, a cost function associated with the current error is deﬁned as Ji =isre f (k + 1) − is (k + 1) (15) where isref is the reference current and is (k + 1) is the predicted current obtained from Equation (14). Ideally, Ji will be equal to its minimum value of (Jmin = 0 in Figure 4) if the AC-side current is controlled well. 5 Energies 2019, 12, 91 3.2.2. Capacitor Voltage Balancing Based on Equations (10) and (11), Vci (k + 1) can be deduced as Si ( k ) S d ( k ) i s ( k ) Vci (k + 1) = Vci (k ) + Ts (16) C where Vci (k) can be measured in real time. Another cost function for balancing the capacitor voltage of sub modules is given as n Jvc = ∑ Vci (k + 1) − Vcire f (k + 1) (17) i =1 where Vciref (k + 1) is the reference DC capacitor voltage of sub module i (with i between 1 and n), n ∑ Vci (k +1) which can be equal to the average voltage of all cells (given as i=1 n ), and Vci (k + 1) is a predicted value, which can be obtained from Equation (16). Consequently, by adding the above cost function together a combined cost function, which can simultaneously achieve the two main control objectives mentioned above is given as the linear combination Jall = α Ji + β Jvc (18) where α and β are weighing factors, α is adjusted based on the cost contribution allocated to the error of AC-Side current, and β is adjusted based on the cost contribution allocated to the voltage deviations of sub module capacitors. The empirical method to determine the value of cost function is presented in [24]. Within each sampling and computing period Ts , the combined cost function Jall is re-calculated, and the best switching indicated to the minimum value for Equation (18) will be adopted for the current control cycle. 3.2.3. Director Switch Control As presented in Figure 2a in Section 2.1, the state of director switches S1 –S4 at the next step should depend on the value of VAC . According to Equation (5), the necessary value of VAC at the current step can be expressed as Ls VAC (k + 1) = Risre f (k + 1) + (isre f (k + 1) − is (k )) (19) Ts However, the ﬂuctuation of VAC (k + 1) due to differences between isref (k + 1) and is (k) during zero voltage crossings will lead to high frequency repeated switching of S1 –S4 , resulting in an increase of switching losses. Therefore, a director switch control strategy should be taken considering the need to 1. Add the energy-balancing mode (Sd = 0 in Table 1) in to achieve energy balancing of the stack of H-bridge by exchanging power with DC bus. 2. Avoid repeated switching of the director switches. Replacing is (k) by isref (k), the necessary value of VAC at the current step can be expressed as L VACre f (k + 1) = Risre f (k + 1) + (i (k + 1) − isre f (k)) (20) Ts sre f where isref (k) is the reference value of the current of the current step. Voltage VACref (k + 1), obtained by Equation (20), is a standard sine wave, which can avoid the ﬂuctuation of VAC (k + 1) due to differences between isref (k + 1) and is (k) during zero voltage crossings. Finally, the implementation procedure of the proposed director switch control strategy is summarized in Part I of Figure 4. The schematic diagram of the control system is shown in Figure 3. 6 Energies 2019, 12, 91 Vciref UDC/n Si isref Predictive Vci(i=1,2Ăn) 1-ph Vci(i=1,2Ăn) control FA-MMC is algorithm Sd is Figure 3. Schematic diagram of the control system. Figure 4. Block diagram of the predictive control strategy 4. Simulation Results This section evaluates the performance of the proposed FA-MMC and control method with a simulation. The simulation parameters are given in Table 2. Ă Ă 7 Energies 2019, 12, 91 Table 2. Parameters of the study system of Figure 1b. DC voltage UDC 3000 V Submodule capacitor C 3300 μF Load inductance Ls 3 mH Buffer inductors Lb 0.1 mH Load inductance R 6 Sampling period Ts 100 μs Nominal frequencies f 50 Hz No. of cell in the stack of H-bridge cells 2 4.1. Operating Performance under a Steady-State Condition Figure 5 shows the voltage of the stack of H-bridges cells, the voltage across the director switches S1 –S4 , and the AC output voltage while the load current tracks the reference in steady-state operation. The simulation results are consistent with the working principle of the topology described in Figure 1b of Section 2. The voltage waveforms appear staircased because there are only two cells, while they would more closely resemble a sine curve with an increase in the number of cells. Figure 6 shows that the capacitor voltages in the two cells are averaged well and mostly under the control of MPC in basic operating mode. Further, they get closer to the given value UDC /ncell_FA in energy-balancing mode. Figure 7 shows the director switch control signal of S1 –S4 . It can be seen in Figure 7a that they all operated at a frequency of 100 Hz and achieved zero voltage switching under the director switch control strategy described in Part I of Figure 4. In contrast, when Part I of Figure 4 is removed, the director switch control signal, which is only determined by VAC (k + 1), is shown in Figure 6b. The difference in responses occurs because VACref (k + 1) in Equation (20) is obviously a standard sine wave while the VAC (k + 1) is repeatedly crossing the zero voltage point as shown in Figure 8. This demonstrates the effectiveness of the director switch control strategy. Figure 9 reveals that the relation of the current across Lb (Ib in Figure 9) and the load current (Is in Figure 9) is similar to Equation (11a) in basic operating mode. The current across Lb (Ib in Figure 9) becomes an overlap current that charges or discharges the capacitor of the cells when S1–S4 are all conducting in energy balancing mode. I Iref 1K 0.5K 0K -0.5K -1K 0 0.02 0.04 0.06 0.08 0.1 T ime (s) (a) Figure 5. Cont. 8 Energies 2019, 12, 91 Vac 6K 4K 2K 0K -2K -4K -6K 0 0.02 0.04 0.06 0.08 0.1 T ime (s) (b) Vpc 4K 3K 2K 1K 0K -1K -2K 0 0.02 0.04 0.06 0.08 0.1 Time (s) (c) Von 5K 4K 3K 2K 1K 0K -1K 0 0.02 0.04 0.06 0.08 0.1 T ime (s) (d) Figure 5. Simulation waveform of the single-phase FA-MMC in steady state operation: (a) Load current and reference current; (b) output AC voltage; (c) voltage of the stack of H-bridges; (d) voltage across the director switches S1 –S4 . Vup1 Vup2 1.6K 1.55K 1.5K 1.45K 1.4K 1.35K 0.02 0.04 0.06 0.08 0.1 T ime (s) Figure 6. Capacitor voltages of the cells. 9 Energies 2019, 12, 91 S1 S2 S3 S4 1 0.8 0.6 0.4 0.2 0.08 0.085 0.09 0.095 T ime (s) (a) S1 S2 S3 S4 1 0.8 0.6 0.4 0.2 0.08 0.085 0.09 0.095 0.1 Time (s) (b) Figure 7. Director switch control signal of S1 –S4 : (a) Control signal based on Vacref(t+Ts) ; (b) control signal based on Vac(t + Ts) . Vnext Vref_next 6K 4K 2K 0K -2K -4K -6K 0.08 0.085 0.09 0.095 0.1 Time (s) Figure 8. The waveforms of Vac and Vacref . 10 Energies 2019, 12, 91 Ib Is 500 0 -500 0.08 0.085 0.09 0.095 0.1 T i me (s) Figure 9. The waveforms of Vac and Vacref . 4.2. Operating Performance under a Transient-State Condition To test the dynamic performance, a sudden change in the reference current is set at 0.04 s, and the behavior of the system is shown in Figure 10. It can be seen that the current tracked the reference value well. The time of reference tracking (from 600 A to −600 A) is less than 0.01 ms as shown in Figure 10b, and the output AC voltage waveform is shown in Figure 11. The capacitor voltage is shown in Figure 12. It can be seen that the capacitor voltage of the two sub-module remains balanced after a sudden change of load current. Figure 12b shows that there is a deviation in the beginning, but is averaged well immediately by the predictive control strategy after 1 ms. I Iref 1K 0.5K 0K -0.5K -1K 0 0.02 0.04 0.06 0.08 0.1 T ime (s) (a) I Iref 500 0 -500 0.039 0.0395 0.04 0.0405 0.041 0.0415 T i me (s) (b) Figure 10. Load current for a sudden change: (a) reference current and actual load current; (b) Detail of the reference current and actual load current at the instant. 11 Energies 2019, 12, 91 Vac 5K 0K -5K 0.02 0.04 0.06 0.08 0.1 T ime (s) (a) Vac 5K 0K -5K 0.039 0.04 0.041 0.042 0.043 T ime (s) (b) Figure 11. Output AC voltage waveforms: (a) Output AC voltage ; (b) detail of the output AC voltage at the instant. Vup1 Vup2 1.6K 1.55K 1.5K 1.45K 1.4K 1.35K 0.02 0.04 0.06 0.08 0.1 T ime (s) (a) Vup1 Vup2 1.55K 1.5K 1.45K 1.4K 0.039 0.04 0.041 0.042 0.043 T ime (s) (b) Figure 12. Capacitor voltages of the sub-module. 12 Energies 2019, 12, 91 Figure 13 shows the states of the director switches at the instant of the sudden change of load current, demonstrating that the director switches are controlled well and operated in low-frequency mode. S1 S2 S3 S4 1 0.8 0.6 0.4 0.2 0 0.02 0.03 0.04 0.05 0.06 Time (s) Figure 13. Control signal of S1 –S4. 4.3. Operating Performance under a DC Fault Having veriﬁed the normal operation of the converter, the model was tested under a DC fault. A three-phase model was built, and a DC fault was induced at 0.04 s. The blocking time is set to be 3 ms after the fault current is detected considering the sensor delay time. Figure 14 shows that the voltage of the cell capacitor is kept at 1.5 kV and the AC current follows the given value before 0.04 s. When a DC short-circuit happens at 0.04 s, the direction of current is reversed and the AC side current rises at ﬁrst because during the sensor delay, the capacitors discharge and current ﬂows from the AC side to the DC side. After 3 ms, when the converter station is blocked, the DC and AC side currents gradually reduce to zero along with the charging of the capacitor. Ia Ib Ic 300 200 100 0 -100 -200 -300 0.02 0.03 0.04 0.05 0.06 T i me (s) (a) Idc 500 0 -500 -1000 0.02 0.03 0.04 0.05 0.06 T i me (s) (b) Figure 14. Cont. 13 Energies 2019, 12, 91 Vcap1 Vcap2 Vcap4 Vcap5 Vcap7 Vcap8 2000 1500 1000 500 0 0.02 0.03 0.04 0.05 0.06 T i me (s) (c) Figure 14. Current and voltage simulation waveforms of a DC fault: (a) AC current; (b) DC current; (c) capacitor voltages. 5. Experimental Results Experiments on an FA-MMC-based inverter were also carried out to verify the proposed topology and test the predictive control strategy. The parameters for the experiment are listed in Table 3. A photo of the inverter is shown in Figure 15 and an IGBT is utilized as the power switch. The main control algorithms were implemented in a combination of a DSP and FPGA. The DC-link voltage was obtained via a three-phase autotransformer. Table 3. Experiment parameters. DC voltage UDC 100 V Submodule capacitor C 3300 μF Load inductance Ls 3 mH Buffer inductors Lb 0.1 mH Load inductance R 6 Sampling period Ts 100 μs Nominal frequencies f 50 Hz No. of cell in the stack of H-bridge cells 2 Figure 15. Photo of the modular multilevel converter (MMC)-based inverter for the experiment. 14 Energies 2019, 12, 91 To test the system balance ability, a 100 Ω resistor was shunted to capacitor SM2. The topology worked in this unbalanced condition by appropriately setting the value of the weighting factor β, which is used to balance the capacitor voltage of the two SMs to zero. In this paper, we set the weighting factor α to a ﬁxed value of 50, and set weighting factor β to 0 or 100 to compare the waveforms. Figure 16 shows the capacitor voltages of the two SMs. At ﬁrst, the capacitor voltage of SM1 is lower than SM2, and the ﬂuctuation is larger due to the unbalanced condition. After giving a suitable value to weighting factor β, each cell capacitor voltage is well regulated to their reference value and the ﬂuctuation of the two cells is also the same. Figure 16. weighting factor β’s effects on the capacitor voltages. Figures 17 and 18 show the output current and voltage of this topology, which are both measured during balanced and unbalanced operation. From Figure 18 we can see that the voltage ripple of the two cell capacitors does not affect the current, apparently due to the robustness of the predictive control. When the weighting factor β is set to 100, meaning that the capacitor voltage balance is considered as a control goal, only a slight distortion is introduced into the output current. 15 Energies 2019, 12, 91 Figure 17. The waveform of the output current under balanced and unbalanced conditions. Figure 18. The waveform of the output voltage under balanced and unbalanced conditions. To test the dynamic performance, the behavior of the FA-MMC and corresponding control method for a step in the angle of the reference current is shown in Figure 19. The waveforms show that the voltage changed quickly to drive the current to its new reference value and that the current is well-tracked. Figure 19 also shows that the dynamic capacitor voltage waveform is not inﬂuenced by the step in the angle of the reference current. Figure 20 shows a detailed view of the output voltage and current for a step in the angle of the reference current. The reference tracking of the proposed method that considered the possible switching states adjacent to VAC (k + 1) is fast, because extreme voltage changes are possible. The results are similar to simulation results. 16 Energies 2019, 12, 91 Figure 19. Waveform of the voltage and current for a step in the angle of the reference current. Figure 20. Detail of the output voltage and current for a step in the angle of the reference current. 6. Characteristic Analysis and Comparison with Other Topologies 6.1. DC Fault Blocking Capacity When a DC-side short-circuit happens, with all IGBTs turned off, the director switches and stack of H-bridges behave as a number of uncontrolled diodes connecting with all DC capacitors in the cells 17 Energies 2019, 12, 91 connected in series, as shown in Figure 21. The equivalent capacitor value in Figure 21 can be expressed as Ce = C/ncell_FA (21) where C is the capacitance of the capacitor in each cell and ncell_FA is the number of cells in one phase of the FA-MMC. The AC source charges the equivalent capacitor and inductors (including the Ls , Lb , Ld ) through the DC fault current, thus, limiting the rising rate of the fault current. Consequently, the value of Uce will rise rapidly, and the DC fault current will be blocked. Figure 21. The equivalent circuit of the insulated-gate bipolar transistors (IGBTs) blocking when a DC fault occurs. 6.2. Number of Sub-module and IGBTs Equation (3) shows that the DC bus voltage is lower than the peak value of output AC voltage by 27% in FA-MMC topology. This implies that the voltage rating of the director switches should be at least equal to the peak value of output AC voltage since they have to support higher voltages. Assuming that the maximum allowable working voltage rating of the IGBTs is equal to the voltage rating of the DC capacitors in the sub-modules, the number of sub-modules of the proposed FA-MMC is given by ∧ U AC ncell_FA = (22) URATED where URATED is the voltage rating of the IGBTs. The number of IGBTs of each phase of the FA-MMC is given by ∧ 8U AC n IGBT1 = 4ncell_FA + 4nS = (23) URATED where nS , the number of IGBT in S1 –S4 , is given by ∧ ∧ V MN U AC nS = = (24) URATED URATED Given the same AC output voltage, we also can deduce the DC voltage, number of cells, and IGBTs needed in an AAMC. The relationship between DC and AC voltage magnitudes in an AAMC, which has been derived in [22], can be expressed as π∧ Vdc = V (25) 2 AC 18 Energies 2019, 12, 91 It can be seen from Equation (3) and (25) that the FA-MMC can reduce the DC bus voltage by half with the same AC output voltage and same active/reactive power ﬂow compared with an AAMC. Considering the sum of the sub-module capacitor voltage must be greater than the peak value of the line-to-line voltage to achieve DC current blocking capability, the number of sub-modules of an AAMC can be expressed as √ Λ 3U AC ncell_AA = (26) URATED But, it only has two direct switches, so considering Equation (24) and (26), the number of IGBTs of a single-phase AAMC is given by √ ∧ (4 3 + 2)U AC n IGBT2 = 4ncell_AA + 2nS = (27) URATED To summarize, the number of IGBTs of the FA-MMC is less than that of the AAMC, and the DC voltage and number of sub-modules of the FA-MMC is nearly half those of the AAMC, leading to smaller size, less need for insulation, and lower cost. And the comparison results of the number of IGBTs and sub-module capacitors between FA-MMC, AAMC and various MMC topologies mentioned in the introduction is shown in Table 4. Table 4. Number of semiconductor devices and sub-module capacitors. Topology Number of Sub-Module Capacitor Number of IGBTs Number of Diodes H-MMC N 2N 0 F-MMC N 4N 0 CH-MMC N 2.35 N 0.7 N AAMC 0.34 N 1.8 N 0 FA-MMC 0.2 N 1.6 N 0 6.3. Efﬁciency Analysis To evaluate the power losses of the FA-MMC and AAMC, a simple loss calculation method for module multilevel converter is adopted [25]. And the result is shown in Table 5. To summarize, the losses of FA2MC increases slightly compared with AAMC, but it is still signiﬁcantly lower than other types of MMC topologies. Table 5. Losses calculation results Topology Switching Losses Conduction Losses Total Losses H-MMC 0.29% 0.82% 1.11% F-MMC 0.29% 1.88% 2.18% CH-MMC 0.29% 1.19% 1.48% AAMC 0.16% 0.47% 0.63% FA-MMC 0.16% 0.66% 0.82% 6.4. Comprehensive Comparison with Other Topological Structures According to the above analysis, a comprehensive comparison between the full-bridge MMC, half-bridge MMC, CH-MMC, A2MC, and FA-MMC is summarized in Table 6, where more “+” means the corresponding topology performs better in the corresponding characteristic. It can be seen in Table 6 that the FA-MMC has advantages in several aspects compared with the other topologies. 19 Energies 2019, 12, 91 Table 6. Comprehensive comparison with other various topology Topology Economy Efﬁciency Volume DC Fault Blocking Capacity Demand for Insulation H-MMC +++ +++ + + + F-MMC + + + ++ + CH-MMC ++ ++ + ++ + AAMC ++++ +++++ ++ ++ ++ FA-MMC +++++ ++++ +++ ++ +++ 7. Conclusions In this paper, a FA-MMC topology and its predictive control scheme have been proposed. The effectiveness of the proposed topology and proposed control strategy under various operating conditions are evaluated based on simulation studies in the PowerSIM environment and experiments, and the comparisons with other topological structures are also given. Through the analysis and demonstration mentioned above, the characteristics of this topology and its predictive control strategy are summarized as follows: (1) The sub-module capacitor number of FA-MMC reduce signiﬁcantly while connecting to the same AC voltage level and power level, results in a more compact structure; (2) Further, it reduces the number of needed IGBTs while retaining the ability to block a DC-side fault compared with other topologies, so that the cost of the system is reduced; (3) The algorithm the algorithm has been proved to be able to achieve multiple control objectives of FA-MMC simultaneously (i.e., capacitor voltages balancing and ac-side currents control). The developed control strategy also contains a director switch control function so that the director switch maintains operation in low-frequency and zero voltage switching mode. Author Contributions: Conceptualization, J.Z.; methodology, T.W.; software, J.Y.; validation, J.Z., Q.H. and J.Y.; writing—original draft preparation, J.Z. Funding: This work was supported by National Key R&D Program of China (2016YFB0900900), the National Natural Science Fund of China (No. 51607171). Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Glossary of Terms EAC The amount of energy transferred from AC side over half the fundamental period EDC The amount of energy going to DC side over half the fundamental period. VAC Output ac voltage IAC AC current UDC DC voltage of FA MMC Ce Equivalent capacitance while all capacitors in cells connected in series Uce_int Initial voltage of Ce when dc fault blocking ϕ Angular position of AC current ncell_FA Number of cells of FA-MMC each phase URATED The rated voltage of IGBT nIGBT1 The needed number of IGBTs of FA-MMC each phase n The needed number of IGBTs of S1–S4 ncell_AA The needed number of cells of AAMC each phase nIGBT2 The needed number of IGBTs of AAMC each phase Vdc DC voltage of AAMC while the output AC voltage is equal to FA MMC VPB The voltage produced by the stack of H-bridge cells of FA MMC Si Switching function of the ith cell Vci The capacitor voltage of the ith cell 20 Energies 2019, 12, 91 Lb Buffer inductor Ib The current through Lb C The capacitor value of cell Sd Switching function of director switch Ls Filter inductor Is The current through Ls Ts Sampling period Jall , Ji , Jvc Cost function α, β Weighting factor References 1. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 22 energies Article A Harmonic Voltage Injection Based DC-Link Imbalance Compensation Technique for Single-Phase Three-Level Neutral-Point-Clamped (NPC) Inverters Kyoung-Pil Kang 1 , Younghoon Cho 1, *, Myung-Hyo Ryu 2 and Ju-Won Baek 2 1 Department of Electrical Engineering, Konkuk University, Seoul 05029, Korea; [email protected] 2 Industry Application Research Laboratory, Korea Electrotechnology Research Institute, Changwon 123456, Korea; [email protected] (M.-H.R.); [email protected] (J.-W.B.) * Correspondence: [email protected]; Tel.: +82-10-6207-0431 Received: 15 May 2018; Accepted: 11 July 2018; Published: 19 July 2018 Abstract: In three-level neutral-point-clamped (NPC) inverters, the voltage imbalance problem between the upper and lower dc-link capacitors is one of the major concerns. This paper proposed a dc-link capacitor voltage balancing method where a common offset voltage was injected. The offset voltage consists of harmonic components and a voltage difference between the upper and the lower capacitors. Here, both the second-order harmonics and the half-wave of the second-order component were injected to compensate for the unbalanced voltage between the capacitors. In order to show the effectiveness of the proposed voltage injection, the theoretical analyses, simulations, and experimental results are provided. Since the proposed method does not require any hardware modiﬁcations, it can be easily adapted. Both the simulations and the experiments validated that the voltage difference of the dc-link could be effectively reduced with the proposed method. Keywords: neutral-point-clamped (NPC) inverter; dc-link capacitor voltage balance; offset voltage injection; harmonic component 1. Introduction Recently, multilevel power inverters have been popularly employed in many electronic applications [1,2]. For example, solid-state transformers (SST) and dc distribution systems, which are high voltage (HV) or medium voltage (MV) applications, essentially require the use of multilevel topologies [3–7]. In multilevel topologies, three-level neutral-point-clamped (NPC) inverters have been widely used in MV and HV applications. Compared to two-level inverters, three-level NPC inverters have some advantages, as follows. NPC inverters have more output voltage levels than two-level inverters. Therefore, the output voltages of an NPC inverter are more similar to sinusoidal waves than other topologies and NPC inverters have less of a harmonic component on output voltage. Additionally, in NPC inverters, the voltage rating of the switching device can be half of the one used in two-level inverters. In addition, NPC inverters generate relatively less leakage current flowing through the ground paths, so electromagnetic interference (EMI) induced problems are relatively lower than the two-level inverters. However, the NPC inverter has a major drawback associated with the neutral-point voltage located between the upper and the lower dc-link capacitors. The voltage between the positive dc-link rail and the neutral-point should be identical to the voltage across the neutral-point and the negative dc-link rail. Unfortunately, there is a voltage imbalance between the upper and lower capacitors. This voltage imbalance harms the stability of the system, and limits the switching operation of the power stage [8–11]. In order to mitigate the voltage imbalance, many strategies that are based on additional hardware conﬁgurations or control algorithms have been proposed, and have been successfully adapted in some applications [12–25]. In [12,13], additional circuits for Energies 2018, 11, 1886; doi:10.3390/en11071886 23 www.mdpi.com/journal/energies Energies 2018, 11, 1886 dc-link balancing were proposed. Although these methods achieved the dc-link voltage balancing successfully, the increase in the cost and the losses were major defects. To avoid these disadvantages, several modulation techniques for single-phase three-level NPC inverters have been presented in [14–21]. Among these modulation techniques, the carrier-based pulse width modulation (CB-PWM) approaches have been extensively preferred due to their simplicity of implementation. In [14], the offset voltage injection with the zero-sequence component in the reference voltage was presented. The zero-sequence component is calculated at every switching period based on the dc-link link voltage and the grid current. Another type of offset voltage injection method was discussed in [15]. In this paper, the offset voltage with a distribution factor was added into the modulation signal. However, these strategies face difﬁculties in being implemented because they are a burden on the prediction of the line current and the avoidance of nonlinearity in the injection signal. Additionally, the exact parameter information is essential to implement these methods as the algorithms are highly dependent on the system parameters. In [16], a simple signal injection method was proposed to balance out the dc-link capacitor voltages by utilizing the harmonic signal consisting of the dc-link voltage difference and the double frequency of the utility grid. The method can easily be implemented as well as reducing the harmonic distortion in the input current of the NPC inverter. In this paper, the method proposed in [16] was further extended and detailed. In the proposed method, an even harmonic signal was added to the reference signal, which is generated by the current controller. Compared to other harmonic injection methods, the proposed method showed less voltage distortion on the synthesized output voltage. Furthermore, fast voltage balancing performance was obtained with the proposed strategy. A 10-kW single-phase three-level NPC inverter was built and tested. Here, the input grid voltage was 943 V in root mean square (RMS) and the output dc-link voltage was 1.8 kV. To artificially create voltage imbalance conditions, an unbalanced load bank was attached to the individual capacitors in the dc-link. The proposed method was compared with the method suggested in [14] through simulations. The experimental results are presented to validate the effectiveness of the proposed method. This paper is organized as follows. In Section 2, the pole voltage of the NPC inverter is analyzed with the proposed offset voltage injection method. The theoretical analysis of control performance with the offset voltage is discussed in Section 3. Simulations and experimental results with the proposed method are shown in Section 4. Finally, Section 5 concludes this paper. 2. The Operation of the Single-Phase NPC Inverter and Its Neutral Current Figure 1 illustrates a switching leg of the three-level NPC inverter and its conduction states. As shown in Figure 1, the switching leg consists of four switching devices, Qx1 , Qx2 , Qx3 , and Qx4 , two clamping diodes, D1 and D2 , and two dc-link capacitors, CCH and CCL . The pole voltage vx0 has three different levels, VCH , 0, and −VCL according to the values of the switch function Sx during the conduction periods, as shown in Figure 1b–d. All parameters used in this paper are deﬁned in Table 1. iCH iCH iCH iCH Qx1 Qx1 Qx1 Qx1 + ON + + + CCH VCH CCH VCH CCH VCH CCH VCH Qx2 − Qx2 − Qx2 − Qx2 − ON ix x idx ix x idx ix x idx ix x idx 0 0 0 0 Qx3 + Qx3 + Qx3 + Qx3 + CCL VCL CCL VCL CCL VCL CCL VCL − − − ON − Qx4 Qx4 Qx4 Qx4 iCL iCL iCL iCL Sx Sx = 1 Sx = 0 Sx = −1 (a) (b) (c) (d) Figure 1. The switching leg of the three-level neutral-point-clamped (NPC) inverter and its switching states. (a) The circuit structure; (b) the conduction state with Sx = 1; (c) the conduction state with Sx = 0; and (d) the conduction state with Sx = −1. 24 Energies 2018, 11, 1886 Table 1. Nomenclature of the hardware and controller parameters. Parameters Description Parameters Description Qxj Power switch “j” in leg “x”. v x0 Average pole voltage. Dx Clamped diode in leg “x”. vZ * Injection voltage reference. Instantaneous current from leg “x” +; − Triangular carrier signals, a positive ix VC VC to grid. (VC + ); and negative (VC − ) one. Individuals capacitances of dc-link Individuals capacitor voltages of CCH , CCL capacitors, the upper (CH); and the V CH , VCL dc-link capacitors, the upper (CH); lower (CL) one. and the lower (CL) one. Resistive output load; and Additional switch to control dc-link Ro ; Radd qadd additional resistive load. capacitor unbalance circuit. Instantaneous voltage of the Instantaneous phase current of NPC eg ig ; ig * grid utility. inverter; and its reference value. dc-link capacitor voltage and its Instantaneous phase voltage of NPC V DC ; V DC * vg ; vg * reference value. inverter; and its reference value. Instantaneous current of Reference signal of leg “x” and its idx ux0 *; u x0 * clamped diode. average value. ı̄d Average current of clamped diode. K Coefﬁcient of injection voltage Lg Input inductance of NPC inverter. ω Angular frequency of phase voltage. τx Pulse width of leg “x”. δg Phase angle of grid voltage. Tc One switching period. m Modulation index. Switching frequency of Sx Switch conduction state of leg “x”. fsw NPC inverter. Instantaneous pole voltage of leg “x” Offset signal for dc-link vx0 ; vx0 * uoffset * and its reference value. balancing control. When Sx = 1, the upper two devices, Qx1 and Qx2 , are turned on. At this condition, shown in Figure 1b, the output power is supplied by the upper dc-link capacitor CCH , so that vx0 becomes VCH . If Sx is 0, the middle switches, Qx2 and Qx3 , conduct, and the output voltage is also clamped by the clamping diodes. Here, the amplitude of vx0 is 0, and this state is illustrated in Figure 1c. In Figure 1d, Sx is deﬁned as −1, and the lower switching devices, Qx3 and Qx4 , are closed. The output voltage is fed by the lower dc-link capacitor CCL , and vx0 becomes −VCL . 1 VC+ 0 vx* 0 −1 VC− Figure 2. The normalized pole voltage reference and the carrier signals. Figure 2 shows the normalized pole voltage reference u∗x0 and two carrier signals VC+ and VC− . Here, u∗x0 is deﬁned as Equation (1): 2v∗x0 u∗x0 = (1) VCH + VCL where v∗x0 is the reference of the pole voltage. Equation (2) deﬁnes the relationships between the magnitudes of the carrier signals and the normalized pole voltage reference and the values of the switching function. ⎧ ⎧ ⎪ ⎪ ⎨ VC+ ≤ u∗x0 ⎨ S x =1 ⎪ if VC− ≤ u∗x0 ≤ VC+ , then Sx =0 (2) ⎪ ⎩ ⎪ ⎪ u∗x0 ≤ VC− ⎩S =−1 x 25 Energies 2018, 11, 1886 Let us consider that the switching frequency fsw is much higher than the frequency of u∗x0 . By doing so, u∗x0 in the single switching period Tc can be assumed as a constant value. Consequently, the on-time duration of the switch, τ x , is given as: τx = |u∗x0 | Tc (3) By using Equation (3), the average output pole voltage, v x0 , over one switching period is simply written as follows: τx VCH /Tc , u∗x0 ≥ 0 v x0 = (4) −τx VCL /Tc , u∗x0 ≤ 0 If the upper and the lower capacitors have the same voltage, V CH = VCL = 0.5V DC , the pole voltage shown in Equation (4) can be rewritten as: τx VDC v x0 = (5) Tc 2 In order to be placed in the linear modulation range, the following conditions should be satisﬁed. VDC |u∗x0 | ≤ 1, |v x0 | ≤ (6) 2 Figure 3 represents the conﬁguration of the single-phase three-level NPC inverter dealt with in this paper. Here, two three-level switching legs were employed. In each switching leg, the middle points of the clamping diodes were connected to the neutral point of the dc-link. By referring the notations in Figure 3, the pole voltage references of the individual switching legs are written as follows: v∗A0 = v∗g + v∗z (7) v∗B0 = v∗z (8) where v∗g and v∗z are the line-to-line voltage reference and the virtual offset voltage between the switching pole B and the neutral point of the dc-link, respectively. It should be noticed that both v∗A0 and v∗B0 should be operated in the linear modulation region. This means that they should satisfy the conditions expressed in Equation (6). In addition, v∗z should be also placed in the linear modulation region, because v∗z is identical to v∗B0 . From this analysis, the following condition can be derived: −0.5V DC − min(v∗g , 0) ≤ v∗z ≤ 0.5VDC − max(v∗g , 0) (9) QA1 QB1 CCH vCH QA 2 QB 2 Lg i g + A + eg vg iA 0 Ro − − B QA 3 QB 3 CCL vCL QA 4 QB 4 Figure 3. The single-phase NPC inverter topology. 26 Energies 2018, 11, 1886 Equation (9) offers that various virtual offset voltages can be selected with various control purposes. The object of the approach taken in this paper was to balance the dc-link capacitor voltages. To do this, the second-order harmonic injection approach was proposed in this paper. Here, v∗z is selected as Equation (10): 1 v∗z = − v∗g + K (VCH − VCL ) sin (2ωt) (10) 2 where ω and K are the fundamental electrical angular frequency of the grid voltage eg and the injection gain of the second harmonic voltage, respectively. By substituting Equation (10) into Equations (7) and (8), the pole voltages are expressed as follows: 1 ∗ v∗A0 = v + K (VCH − VCL ) sin (2ωt) (11) 2 g 1 ∗ v∗B0 = v + K (VCH − VCL ) sin (2ωt) (12) 2 g In Figure 3, the voltage of the lower capacitor was adjusted by injecting the neutral current idA and idB , which are represented with the phase current from the switching pole A to the grid iA and the switching functions of each switching leg, SA0 and SB0 , as follows. idA (t) = [1 − S2A (t)]i A (t) (13) idB (t) = −[1 − S2B (t)]i A (t) (14) The entire neutral current ﬂowing into the neutral point is simply obtained as: id (t) = idA (t) + idB (t) = [S2B (t) − S2A (t)]i A (t) (15) The average value of the neutral current over a single switching period is calculated as below: 1 id = id (t)dt = (|u∗B0 | − |u∗A0 |)i A (16) Tc Tc where i A is the average of iA (t) in the switching period. By applying the pole voltages, Equation (16) is rewritten as Equation (17). 2 id = (|v∗ | − |v∗A0 |)i A (17) VDC B0 By substituting Equations (11) and (12) into Equation (17), id is obtained, and is expressed in two ways according to the polarities of v∗A0 and v∗B0 . When the polarities of the pole voltage references are the same, the average neutral current is written as Equation (18). 2v∗g id = ∓ (18) VDC If the polarities of v∗A0 and v∗B0 are different, Equation (19) is obtained. 4K id = ∓ sin(2ωt) (19) VDC By adjusting the pole voltage references, the average current expressed by Equations (18) or (19) is controlled to balance out the dc-link capacitors’ voltages. 27